E398 
.L47 



^?:^-" .0*^' 






^s^:^,- ^(^ 



.^^,- 






>P^^^ 









. v ->■ 






A 






O > 

4. V--^v;\ir^_ 












V 









o 



v<i^' 



•* 

^<^^ 

♦^ 



■ '^'5' ^^ */ 

.0' - ■ ' o 












^ 

V 



V 



v^ 



4 o 












II 



^5 J^ 






-^^0^ 






/X^^-v^V-^^"^ 










r-^^. 



.V W^f'T*,," 



ok/ 



c^^ 



-I 1^ - 



*^ 






A LETTER 

TO THE 

PRESIDENT OF THE U. STATES OF AMERICA; 

PROPOSING A METHOD WHEREBY THE MERITS OF 
TEIE CONFLICTING CLAIMS OF 

THE UNITED STATES AND GREAT BRITAIN, 
ON THE DISPUTED FRONTIER, 

MAV BE CORRECTLY EXAMINED AXD DETERMINED, BY SCIENTIFIC 

PRINCIPLES, COMBINED WITH IND ISP LTABLE DATA, 

GROUNDED IN EQUITY; 

TOGETHER WITH AN EXAMINATION OF 

THE TREATY OF 17S3: 

FROM WHICH IS DERIVED A DEMONSTRATION OF THE FALSEHOOD OF 

THE BRITISH CL\IM. 



By JOHN LEE, 

DECYPHEREE OF THE CHARTER OF OMAGB, AND AUTHOR OF PROPORTIONAL 

FORMULA, THEORY OF LONGITODE, AND A TREATISE ON THE 

ERRORS OF THE RECTANGULAR SURVEY. 



' " A}./io} S" iv ari'i&taai ri&ti voov h^qvottu itvf 

Ea9'/iMV, Tou a rt noXXoi tTiavQiaxoiT^ oit-9Qw7tot, 
Kal TE TTo/lfi; iauwae." — Iliad, Lib. XIII. 

" I, wisdom, dwell with prudence, and find out knowledge of witty inven- 
tions." — Proverbs, viii. 12. 

<' Wisdom is better than weapons of war." — Ecclesiastes, ix. 18. 



CAIil BRIDGE: 

METCALF, TORRY, AND BALLOU. 

183 9. 






Entered, according to act of Congress, in the year 1839, by Metcalf, 
ToRRY, AND Ballou, In the Clerk's Office of the District Court of the 
District of Massachusetts. 



A LETTER 



TO HIS EXCELLENCY THE 



PRESIDENT OF THE UNITED STATES OF AMERICA. 

Sir: 

The conspicuous majesty of Science is now recognised 
from nation to nation, and from clime to clime, throngli the 
circuit of the habitable universe. From her intellectual 
throne she swaya the destinies of men. In the remote and 
tranquil scenes of rural cultivation; in the busy throng of 
traffic and the ceaseless reverberating din which pervades the 
commercial cities ; on the vast and solitary surface of the 
undulating deep; in the barriers of eternal polar ice, or the 
impenetrable silent horrors of subterrestrial gloom ; through 
all the confines of the sublunary world, she claims ubiquity 
of empire. That portion of our being, which we inherit from 
divinity, has been, by the discipline of Science, developed in 
such magnificence of energy as clearly marks its origin ; 
while the petty powers of our animal constitution have sunk 
to comparative nonentity. In the conflict of battle, and the 
agonizing shock of war, the achievements of corporeal prow- 
ess are behold no more; the mail-protected rank of heroes has 
dwindled into puny insignificance, before the vast machinery 
of destruction which Science has created and arrayed upon 
the martial field; she impels, with invisible and superhuman 
arm, the flame-winged missiles which annihilate legions, and 
crush the pride of castellated bulwarks with irresistible and 
ruinous prostration. 

This development of the mental energy of man, though its 
recent progress to maturity be characterized with gigantic and 



marvellous rapidity, is no precipitate result of late or sudden 
causes. It is the gradual growth and final fructification of 
the tree of Knowledge, through the total extent of a period 
Mdiich has nearly comprehended six thousand years. Through 
the reckless and barbarian atheism of lawless nnconfederaied 
man ; through the mystic incantations and fantastic horrors 
which mark the fearful rites of idolatrous bewildered nations; 
and in later times, through all the seductive sophistry and 
blasphemous impiety of audacious and perverse philosophy ; 
like the vegetative bud which survives the changes of the 
atmosphere, the intellectual germ has resisted every ungenial 
influence of place and time, and preserved its undecayed vi- 
tality, to attain the plenitude and lustre of autumnal exuber- 
ance and bloom. 

And, tlierefore, when contrasted with the flaming splendor 
which illuminates the annals of science in the present age, 
the page oi ancient history is not completely ^esolaie. and dark. 
Through the mist of intervening ages, the achievements of 
the peerless Archimedes present a monumental prototype of 
almost every marvellous phenomenon, which, by the aid of 
science, is created on the theatre of nations in the present day. 
His caustic apparatus and projectile engines refer the imagin- 
ation, by a simultaneous and triple analogy, to the sudden 
flash and heavy roar of the cannon, to the lightning and 
thunder of the electric machine, and the irresistible energy 
which results from the concentration of the optic ray. His 
investigations in the theory of motion, equilibrium, and im- 
pulse, have served as- a basis for all that we now know in 
mechanical science, through all its modifications and depart- 
ments, comprehending the multitudinous motive powers and 
impulsive agencies of solid and fluid materials. And in bright 
and glorious addition to all the preceding discoveries, his dar- 
ing and sublime excursions, in the intellectual region of pure 
Geometry, through unexplored and solitary tracts of knowl- 
edge, but copious and luxuriant in the intrinsic and immortal 
beauty of the immutable nature of things ; these intrepid 
fiio'hts of reason have directed the research of every subse- 
quent adventurer on that boundless and eternal field. 



Sir, it is not necessary here, to enumerate, in detail, the im- 
portant services for wliich celebrity is due to otlier numerous 
and illustrious names, which adorn the recorded catalogue of 
scientific labors and lore, through the long and shadowy tract 
of time comprehended in ancient history ; though the mental 
faculties of your Excellency must have been, for years, indis- 
pensably pre-occupied with all the responsibilities and cares 
attached to the superintendence of the political alfairs of a great 
and rising nation; I consider it yet a presumable circumstance, 
that all the classic recollections, which refer to the develop- 
ment and progress of the human mind from early imbecility 
to intellectual manhood, have not completely faded from your 
memory. 

Though the extraordinary and startling problem, the per- 
formance of which is the professed and immediate object of 
the present letter, be one which is indeed unexampled in the 
history of science, the preceding rapid sketch of the continu- 
ous movements and incessant researches of the human mind, 
will, I am disposed to believe, diminish the deep surprise, 
with which, for the first time, your Excellency, and the public 
wouldotherwise receive the intimation of theattcmpted solulion 
of this problem. By a simple and direct pursuit of the ideal 
train of associations presented in the preceding sketch, the 
solution of this problem will easily apj)ear to he part and par- 
cel of the connected results of the unquiet and aspiring ope- 
rations of the general human mind ; of that primeval, 
celestial, and elastic impulse, whereby our universal race and 
lineage are exalted above the humility of their native dust, to 
tread, in intellectual pre-eminence, the ulterior confines of the 
vaulted universe, and contribute recognition and homage to 
the invisible Architect who dwells beyond. 

Sir, I am a native and voluntary subject of the British 
sovereign ; but iew and doubtful are the claims of gratitude, 
however, which the British nation, or any other, has on my 
individual affections; this country I have sought, not as a 
refuge for political disaffection, or a laboratory for the experi- 
mental career of a political empiric; not as the ultimate re- 
source of one who escapes from the horrors of penury, nor 



yet as a protecting asylum for an infamous and guilty fugitive ; 
I have directed my steps upon the transatlantic shore, to 
obtain a refuge, in retirement, from scenes of adversity, and 
a cessation of the miseries of mental distress ; to seek a rest- 
ing place where sources of private unhappiness may feel the 
balm of solitude ; where the eyes of the depraved rabble may 
never enjoy that luxury which is deemed intensely grateful to 
malignity and envy, when permitted to espy the faded lustre 
of decayed respectability, and the ruinous operations of disas- 
trous and reverted fortune ; where the afflicted proud man, 
when he feels the burning and resistless impulse to relieve and 
vent his anguish by external or internal tears, may weep in 
secret and alone. 

But while the self-dependent and secluded nature of my 
present condition compels me to regard my fealty as my own, 
and brook no infringement of my personal rights from any 
existing human power, when the means of retaliation are ac- 
cessible ; I cling to the belief, that I feel myself too deeply 
bound by the obligations of moral conscience, by the intrinsic 
sanctity of justice, by the love of equity which dwells immu- 
tably inherent in the virtuous mind, and by the lingering at- 
tachments of pre-existing national sympathy which distinguish 
and adorn the human character, and interpose a true criterion 
betwixt the magnanimous and the servile spirit ; I believe 
myself too potently restricted by these considerations, to es- 
pouse the American claims for the sake of popular favor, 
in considering the merits of the great international dispute 
which I propose to examine in this letter. 

I must also observe to your Excellency, that, while I con- 
ceive myself, in this country, deeply indebted to a few indi- 
vidual friends; yet, from the nation at large, or any section, 
great or small, of the community, I derive no favor, and there- 
fore acknowledge no obligation ; I anticipate no advantage, and 
seek no benefit, except as the reward of my own exertions; 
and few therefore, I presume, can be easily found, who are 
better qualified by total exemption from ''-fear, favor, or affec- 
tion,'' than I am, for the examination of this great question. 

Having offered the preceding remarks to your Excellency, 



for the vindication of my own character, conduct, and motives, 
m attempting to determine the merits and dispel the perplexi- 
ties of this dispute, I now proceed, without further preface 
or delay, to enter on the field of explanation and decision. 
And here, in the audible and free communication, and the 
open conspicuous prospect of the great and general amphithea- 
tre of nations, I avow myself constrained in justice, to give 
my direct and unequivocal declaration and opinion on the 
side of the American party in this litigation. I am constrained 
to acknowledge, that every feature of the British claim so 
visibly and palpably betrays the stamp of artificial and recent 
origin, as to seal the lips of every conscientious and enlighten- 
ed man who attempts to speak in its deience. I am constrain- 
ed to deprecate, with horror and disdain, the dark, the damna- 
ble, perverse, and baleful doctrine of misguided ethical 
philosophy, which tolerates an act of iniquity and fraud, if 
such act be the act of a nation ; which designates extortion by 
the title of refined policy, if a nation, as one man, have deter- 
mined to act in concert as unanimous confederate extortioners ; 
which covers political villainy with gaudy epithets, and hides 
the infamy and shame of a people in the frivolous delusions of 
insignificant verbal vanity ; which encourages the national 
and public perpetration of deeds, the doom and consequence 
of which, to the performer, if performed by one individual, 
would eject him, as a vile, abhorred, and solitary fugitive, 
abroad upon the face of the earth. I am constrained to ac- 
knowledge, that if the lion be correctly taken for a symbol of 
the nobler characteristics of the British nation, the same anal- 
ogy also too unhappily and closely prevails, in respect of some 
other less attractive traits of character which mark the royal 
quadruped ; I am constrained to express my fear, that, if con- 
sidered in reference to such a comparison, the acts of that 
nation have more than once betrayed a propensity — royal in 
cupidity, and more than royal in rapacity. 

In the examination of the question of the frontier, which I 
propose to undertake, on scientific principles, in the subsequent 
part of this letter, many mathematical theorems are involved, 
of so technical a nature, that I should be guilty of gross and 



contemptible hypocrisy, by pretending to presume that all 
those theorems are familiar to your Excellency, exercised and 
occupied as you have been for many years, in the responsibili- 
ties and duties of political affairs. For the satisfaction of 
your Excellency concerning those technical inquiries, I pre- 
sume to suggest a reference, if necessary, to the mathematical 
authorities of Yale, Virginia, and West Point. 

Finally, Sir, before I undertake the proposed examination, 
I must express the hope, that a consideration of the nature of 
my attempt will tend, with additional force, to impress upon 
your Excellency, a deep conviction of the incalculable benefits 
resulting to maiikind, in every conceivable department of 
human affairs, from the patronage, diffusion, and protection of 
science. I shall rejoice, if enabled hereafter to believe, that, by 
any observation contained in or suggested by this letter, the 
attention of your Excellency, or that of any other influential 
man, has been more closely directed to the promotion of that 
glorious object. 

But I cannot incur the deep culpability of neglecting to 
inform your Excellency, and all my fellow-men, that mathe- 
matical science, like every other human pursuit, has its vani- 
ties, deceits, and snares. More especially in later times, a 
race of men has appeared upon the field of demonstration, 
the devious meteoric aspect of whose intellectual career has 
induced me to apply to such indviduals the epithet, Formu- 
larian, as a distinctive afjpellation. These men shun defini- 
tions, and refuse to explore first principles ; they conceal am- 
biguity by apparently precise, but latently equivocal words; 
they accomplish, on formulcp, unusual and curious transform- 
ations, by unjustly, though plausibly, gcnercdizing the 
ordinary rides of opei^ation, like certain other philosophers, 
"who, having once discovered that a circumstance is very 
generally true, immediately arm themselves with such a 
discovery, as a weapon wherewith to deiiy the reality of every 
case of exception which may afterwards occur. The absurdi- 
ties of these men are sometimes harmless and amusing ; but 
unhappily, too often, more calamitous results ensue. Among 
this visionary tribe, La Place appears to be pre-eminent, in 



propensity to mischief, and in power of performance. The 
splendid fictions and marvellous delusions of that unrivalled 
Formidarian are pregnant with destruction ; they are sub- 
versive, in their final tendency, of religion, morality, and 
social order; and consequently hostile to the present and sub- 
sequent happiness of men. Such individuals may be rightly 
regarded as the despicable vermin which pollute, while they 
prey upon, the intellectual decayed remains of Archimedes 
and of Newton. 

Before I undertake the proposed question, I shall finally 
observe to your Excellency, and the public, that, concerning 
the promotion of science, I am sorry to perceive the existence 
of an error, which, in politics and literature, is equally fatal 
and disastrous. This error consists in the supposed utiliti/ of 
domineering behavior. One of the predecessors of your 
Excellency, in the exalted station which you now occupy, 
unfortunately tarnished, by a fatal error of that nature, the 
lustre of his previous reputation ; but how much more inde- 
scribably contemptible is the aspect of a domineering bully 
among juvenile students^ than that of a rash and arbitrary 
man directing the affairs of a nation ! The only considera- 
tion, that operates with resistless and perpetual sway on every 
department and rank of human society, is the prepossessing 
native dignity icldch peculiarly discriminates the gentleman ; 
and which is completely incompatible with every vestige of 
that repulsive rudeness, which betrays the barbarian, the 
ruffian, or the bear. 

Finally, I now proceed, according to proposal, to furnish 
in detail, to your Excellency, the scientific investigation and 
solution of the problem respecting the disputed frontier. 



10 



INVESTIGATION OF THE PROBLEM OF THE DISPUTED 

FRONTIER. 

Art. 1. If two parties agree to discriminate a tract of 
country into two several shares, by a transverse boundary, and 
the face of the country present no natural obstacles, that 
boundary will, beyond all doubt, be Rectilineal; for every 
curvilinear deflection or angular deviation would be a source 
of improfitable toil and useless perplexity. 

Art. 2. And hence if ixny part of such boundary were 
disputed or defaced, it might be, at any time, re-ascertained, 
by simply pursuing the direction of that part which is known. 

Art. 3. But if natural obstacles occur on the face of the 
country, they will operate in two ways ; first, by intercept- 
ing the course and precluding the progress of the intended 
rectilineal boundary ; and secondly, by presenting, in them- 
selves, a range of immutable and definite local features, which 
may, with facility and certainty, be taken as connecting-points 
through which we may imagine a boundary to pass. 

Art. 4. Between the parties, however, a previous under- 
standing must have existed, either strict, precise, and immu- 
table, or else restrained within certain limits of allowable 
adjustment, concerning the proportionate magnitudes and 
relative situations of the two shares. 

Art. 5. And hence, in the selection of such natural con- 
necting-points, to mark the direction of a boundary, the pro- 
ceedings will be governed and affected by this indispensable 
consideration ; so to direct such a boundary, as to violate in the 
smallest possible degree, the understanding which existed, 
respecting tlie situations and magnitudes of the two shares. 

Art. 6. And if the series of natural points discontinue, 
before this boundary has completed the extent of its course, 
and the said boundary, having reached the last of these points, 
emerge upon the uniform face of the country, so that, beyond 
that point, its direction be optional; the only consideration 
thenceforth existing, which can regulate the course of that 
final part of the boundary, is this : so to direct this final part, 



11 

as to compensate for any violation of the aforesaid previous 
ujiderstanding, which may, of necessity, have arisen, by adopt- 
ing- this train of natural points, to regidate the course of the 
preceding part. 

Art. 7. And this consideration is equally important and 
decisive, at either extremity of the said boundary, or at both ; 
the term ^^final,'^ in the previous article, being merely adopted 
for facility of expression ; since, of the two extremities of any 
such boundary, either may be taken as the initial, and the 
other as the final extremity. 

Art. 8. The import of Articles 6, 6, and 7, connectively 
taken, is expressible thus. In the prescription of any such 
boundary, the total course of the proceedings will be govern- 
ed by this consideration ; .so to accommodate the several parts 
of that boundary to each other, as to give to the total boundary, 
precisely, or as nearly as possible, a self-adjusting character ; 
that is, that all infringements of the general prc-under stood 
conditions, which are unavoidably made in certain parts of 
that boundary, in favor of one party, are counteracted and 
recompensed by other infringements which are elsewhere made 
in the course of that boundary, in favor of the other party. 

Art. 9. Now, as we have observed in Article 1, that all 
])oundaries, if not intercepted by obstacles, would be rectilin- 
eal ; and as all desirable conditions, concerning the relative 
situations and magnitudes of two shares, can be satisfied by 
merely accommodating the situation and direction of a rec- 
tilineal boundary to each particular case; it fdllov/s, that, 
whenever the separation and distinction of two several shares 
is required, a straight line might be drawn somewhere, which 
would satisfy the general pre-understood conditions between 
the parties. Such a straight line we shall call a Normal. 

Art. 10. From Arts. 8 and 9 we learn, that if any bounda- 
ry whatever between two shares be completely self-adjusted, 
the following remarkable relation must be verified between it, 
and evej-y normal ichatever, appertaining to the same case 
between the same two parties. In Fig. 1, let N n be a nor- 
mal, and ABCDEFG be a self-adjusted boundary, Q q and 
R r being Qxterior boundaries of the total tract which com- 



12 

prehends the two shares ; then, the sum of all the areas, 
a, (i, y, s, &c., which are mtercepted by A^ n, and those parts 
of the tortuous boundary which lie on one side of it, is equal 
to the sum of a, b, c, d, &c., which are intercepted by A'' n, 
and by those parts of the tortuous line which lie on the other 
side of it. For, if the two sums be unequal, the straight and 
the tortuous boundary cannot both satisfy the understood con- 
ditions of the relative magnitudes of the two shares ; since, 
by the adoption of one boundary, one party has a greater 
share, and tiie other a less, than by the adoption of the other 
boundary ; but both boundaries do satisfy the aforesaid condi- 
tions ; A^ w, by its character as a noi^mal, and the tortuous 
boundary by self-adjustment ; therefore the two sums must 
be equal. Q, . E . D. 

Art. 11. Whenever a straight line bears to any other line 
such a relation, with respect to the intercepted areas, as that 
which N n bears to the tortuous boundary in Art. 10, we 
shall call that straight line, a coequator to the other line. 

Art. 12. But yet, among tortuous lines, the general bear- 
ing or progressive tendency of one, may depart from that of 
its coequator, much more than that of another tortuous line 
from its coequator. Thus, in Figs. 2 and 3, the general bear- 
ings of the tortuous lines ABCDEF, and a b c d e f, exactly 
coincide in direction with their coequators ^F and af, Avhile 
the general bearing of a b c d e fg h, in Fig. 4, is itself the 
tortuous line a d i, intersecting the coequator a 7c, AB and 
CD being the exterior boundaries comprehending the total 
tract which contains the two shares. Whenever the coequa- 
tor and the general bearing exactly coincide, the two extremi- 
ties of the coequator and those of the tortuous line will also 
exactly coincide, as evidently appears by Figs, 2 and 3. This 
relation we shall express by saying, that, in every such case, 
the tortuous line preserves perfect affinity with its coequator ; 
but in other cases, the want of such affinity shall be express- 
ed by stating, that it more or less diverges from that coequator. 

Art. 13. Before proceeding further, it is well to consider, 
in what circumstance this greater or less divergency cojisists. 
Two straight lines, which coincide in direction, make no an- 



13 

gle. Now in Figs. 2 and 3, the coequator AF, in the former 
case, and a f, in the latter, coincides, or is identical with, a 
straight line uniting the extremities of the boundary to which it 
is draivn as a coequator. In each case, the coequator makes no 
angle with the connecting Hne, and therefore we say that such 
boundary has no divergence from that coequator, or has perfect 
affinity with it, by which we mean the same thing. But, in 
Fig. 4, the straight line a h, which connects the extremities 
of the dotted boundary, does make an angle with the coequa- 
tor a Jc, and therefore we say that such boundary diverges 
from that coequator. Lastly, in Fig. 5, if a b c d he a bounda- 
ry, and a d, ef and ^'^ h, be all coequators, AB and CD being 
the exterior boundaries, comprehending the total tract which 
contains the two shares; we perceive that abed has no di- 
vergence from a d, but it has a divergence from ef, as we per- 
ceive by the angles " and (f, which ef makes with a d. But 
abed has a yet greater divergence from g h ; for g h makes 
angles with a d, which severally exceed the two former, by 
the angles '/ and •'■ 

Art. 14. But here an indispensable remark must be made. 
For, since any two straight lines, which meet, will make, if 
produced, as in Fig. Q, four a7iglcs equal by pairs, as a and «, b 
and 1^' and since, if either pair be very small, the divergence of 
the two lilies will be very small ; therefore, of two adjacent 
angles, which one straight line makes with another, as the 
two angles a and b, in Fig 7 ; if these two be unequal, the 
minor angle is that whereby we measure the divergence. 

AiiT. 15. Now as it appears from Arts. I and 9, that all 
boundaries, if not encountered by natural impediments, would 
be Norrnals, because boundaries of such a character offer to the 
human understanding the only obvious, primary, and imme- 
diate resource whereby to realize the pre-conceived conditions 
between the parties ; it follows, that where a boundary has 
been eventually adopted between two such parties, the frst 
notion of such boundary must have been that of a noimal^ 
which, for the sake of accommodation to existing circum- 
stances, has been diversified and altered into that form under 
which it was finally adopted. In every such case, the pre- 



14 

conceived original normal we shall call the Archetype, and the 
actual boundary therefrom derived we shall call the Metatype. 

Art. 16. Hence we obtain a test for the detection of a spu- 
rious boundary. For if we can collect any evidence to show 
that such boundary has been obtained and derived from no 
normal archetype, we have established a proof, that such boini- 
dary was never adopted as the result of the mutual delibera- 
tion of two contracting parties, but betrays a fictitious origin 
by the devious and distorted aspect of its general career ; pre- 
senting, indeed, no unsuitable picture of the inconsistent, 
perverse, and guilty movements of the creating spirit which 
directed it ; alternately impelled and restricted by the stimu- 
lations of encroaching avarice, and the retiring trepidation of 
abashed and conscious turpitude. 

Art. 17. To all that feel disposed, without reflection, to 
denounce the principle detailed in the preceding Article, as 
far-fetched and fanciful, it may be well to observe, that «/" that 
principle be a mere metaphysical and idle conceit, the fact is 
very extraordinary, that, from the most primitive and unspec- 
ulating ages of the human race, down to the present time, the 
universal structure of language will clearly prove, that, from 
the geometrical ideas of " Siraight,^^ and ^^ Crooked,^' ^'Even," 
and " Uneven,^^ the moral ideas of ^^Good,''^ and "^i;«7," have, 
in virtue of the right of kindred, borrowed their verbal habil- 
iments. With moderate research, and possibly with some sur- 
prise, those objectors will discover, that ^'Equity,'' is ''Flat- 
ness,^' and moral Mectitude is moral Straightness. They will 
also perceive that a ''Delinquent^' is one who siverves from 
the path of duty, so as to " Leave"" that path lying ''Off" or 
away from his present course. They will also further perceive, 
that a "Perverse" disposition is that which is "Very Much 
Twisted Away." To multi})ly examples of this kind is not 
necessary ; they are familiar to every linguist, and clearly in- 
dicate an intrinsic similarity and a usual association of these 
geometrical and moral ideas, for which no satisfactory reason 
can be easily given, if we deny the proposition, that a tendency 
to straightness will be a prevailing feature in a boundary 
traced for the first time between two tracts of land, under the 
direction of honest men. 



Art. 18. The process of deriving a boundary irom its nor- 
mal archetype will be evidently guided and ruled by this con- 
sideration ; to shun mis-regulation, perplexity, and toil, hy di- 
recting this boundary, so as, first, to render it self-adjusted, 
whereby the normal archetype will be its coequator ; a7id sc- 
condly, so as, if possible, to have perfect affinity with its nor- 
mal archetype, or else to diverge from that archetype in the 
smallest possible degree. A familiar illustration of this case 
may be drawn from that of a traveller, who, being led by an- 
other person through an unknown country, will deviate from 
the trade of his guide as little as possible. 

Art. 19. Hence we obtain a satisfactory test whereby we 
may discover, at least in extreme cases, whether any boundary 
has been derived from a normal archetype, or fabricated on some 
other principles. For if we can discover that such boundary 
diverges extravagantly from each of all its possible coequators, 
wc have clearly a strong presumption that such boundary ivas 
not obtained from any normal archetype ; and, on the other hand, 
if we discover that such boundary has 07ie coequator to which 
it has perfect affinity, or from which it very slightly diverges, we 
have an equally valid presumption of the opposite kind. 

Art. 20. But a question may occur, of the same general na- 
ture, but under a diilerent form, wherein the preceding test 
may be safely applied, not only in extreme cases, but in any 
case. For if two boundaries be placed, from the peculiar 
cause, which gave them origin, or from any other circum- 
stance, under such conflicting conditions and relations, one to 
the other, that one of these boundaries 7nust have been derived 
from a normal archetype, and the other must have been fabricated 
on other principles ; and if we ascertain that we can draw 
to one of these boundaries a coequator to which it has perfect 
affinity, whereas none such can be drawn to the other; or else, 
if we draw to each boundary that coequator from which it 
has least divergence ; and then discover the divergence of the 
former boundary from such coequator to be less than that of 
the latter from its own coequator ; we obtain thus, for the for- 
m.er boundary, a cause of preference before the latter, which de- 
cides the question. 



16 

Art. 21. It is also observable that sometimes a coequator 
may be parallel to the straight Hne which connects the ex- 
tremities of the boundary, as de to ac, which connects the ex- 
tremities of the boundary ah c^ in Fig, 8 ; or, though not par- 
allel, may not meet that straight line, unless both be produced 
beyond the total tract, as the coequator/^, in Fig. 9, meets a e 
in g, the exterior borders of the tract being ABCDE and 
I GHI. All such coequators we shall designate, for the sake 
of distinction, as Remote, and all others as Adjacent. From the 
extreme difficulty, however, of directing an adjusted boundary 
by any such remote coequator, it is needless to say that no 
boundary has ever been derived from such a coequator as its 
normal archetype. In the examination, therefore, of the com- 
parative claims of difierent boundaries, to a genuine origin 
from a normal archetype, we have no cause to make any in- 
quiry with respect to remote coequators. 

Art. 22. If two boundaries be so circumstanced, that ojie 
must be spurious and the other genuine, but — which to ac- 
knowledge as genimie, — and which to reject — we are uncer- 
tain ; the only necessary aid, which has not yet been furnished 
for the settlement of such a question, is a process whereby we 
can try the question of greater or less divergency, as stated in 
the preceding Articles. For, by such a process, we can pre- 
sumptively ascertain, as appears by those Articles, which of 
those boundaries has been derived from a normal archetype, 
and which was fabricated on other principles ; and again by 
Art. 16, from the discovery of these latter facts, we are imme- 
diately enabled to discriminate the spurious from the gen- 
uine boundary. 

Art. 23. From each of the two extremities of any bounda- 
ry a coequator to that boundary may be drawn. For, let a?iy 
straight line, as dein Fig. 10 and 11, be drawn from one ex; 
tremity, as d, of a boundary abed, till it meets the opposite 
exterior border of the tract in e; de either not again meeting 
the boundary, as in Fig. 10, or again meeting it one or more 
times, as in Fig. 11. In the former case, let the area of the 
space comprehended between the boundary abed, the straight 
line d e, and the border a e, be calculated and represented by 



17 

A ; it is now an easy geodesic problem, to determine the posi- 
tion of a straight line, as the doited line da, which will make 
the space comprehended between itself, de, and the border, 
equal to A. The straight line d a is the required coequator. 
For since d c augments one of the shares, as determined by the 
boundary, by the quantity A ; and again d a diminishes by the 
same quantity A, the share which has thus been augmented ; 
that share, and consequently the opposite one also, is now re- 
stored to its former value ; that is, da is a coequator. In the 
latter case, let all the areas intercepted by d e, by the bounda- 
ry, and by the opposite border, on one side of de, be collected 
into one sum ; and let all such areas on the o^/tcr side of de he 
collected and summed in the same manner ; and if, on one 
side, no intercepted area be found, the sum of such areas on 
that side may be stated as equal to zero ; take the difference of 
the two sums, and denote it by t^; then, upon that side of de, 
where the sum of the intercepted areas is greater than the 
other, draw df so as to make the space comprehended be- 
tween itself, de, and the border, equal to (5; then for the same 
reasons which were assigned in the preceding case, dfmust 
be a coequator. Every coequator, which meets one of the ex- 
tremities of a boundary, shall hereafter, for the sake of distinc- 
tion, be styled Conterminal, and every other coequator, Dis- 
terminal. 

Art. 24. All coequators to the same boundary must meet 
and intersect within the exterior borders of the total tract 
which contains the two shares. For if two straight lines meet 
on such exterior border, as a c and b c. Fig. 12, which meet on 
the border CD ; or if they meet neither on nor within such bor- 
der, as a c and h d. Fig. 13 ; then one of the two shares, deter- 
mined by one of these straight lines, is augmented by the other 
straight line, without any counteracting diminution ; therefore 
they cannot be both coequators. Therefore any two coequa- 
tors must intersect within the borders, as a i and c d, Fig. 14. 

Art. 25. And hence it immediately results, that, from either 
extremity of a boundary, we can draw but one coequator; 
3 



18 

tiiat IS, to cyeiy boundary, we can draw tioo, and only tw 
conterminal coequators ; and, in the case of perfect affinity, these 
two coincide in direction, and thus, in a certain sense, become 
one. 

Art. 26. Of all coequators drawn to the same boundary, 
remote ones excepted, one of the two conterminal coequators 
is that which has least divergence. For, in Fig. 15, let a and 
b be the two extremities of a boundary, which it is not neces- 
sary here to represent ; let a c and b d be the two conterminal 
coequators, intersecting in g ; let e Tc be any other coequator 
which is not remote, intersecting a c in h, and b d in i ; be- 
cause e it is not remote, it meets the connecting line a b, at 
some point/, between a and b; now of the triangle i/ i, the 
exterior angle « is greater than the angle at b; and of the tri- 
angle afhthe ^exterior angle ^ is greater than the angle at a ; 
therefore, of « and /?, the minor one, if these be unequal, or 
each of them, if equal, is greater than one of the two interior 
angles at a and b ; that is, in any case ; one of the two interior 
angles at a and b, is less than each of the angles at « and /s. 
Of that interior angle and its adjacent exterior, if the interior 
be the minor angle, or if both be equal, the conterminal coe- 
quator, to which that angle appertains, must have less diver- 
gence than e h: ; but if the adjacent exterior angle be the mi- 
nor, then, since the interior angle is less than each of the two 
angles « and ,-?, a fortiori, the exterior angle, which is now the 
minor, is less than each of the angles a and P ; therefore, in 
every possible case, one of the two conterminal coequators has 
less divergence than e Tc. Q. E. D. 

Art, 27. Hence, if the conflicting claims of two boundaries 
be such, that one must be spurious and the other genuine, we 
have the following practical rule for a decision -oi the case. 
To one boundary, draw both its conterminal coequators, and ascer- 
tain the divergence of each, selecting the lesser divergence, or ei- 
ther, if equal, as the least possible which that boundary can have 
with any coequator which is not remote ; in like manner, discover 
the least divergence, in the case of the other boundary; compare, 
these two results together ; then, whichever boundary affords the. 



19 

less result, we obtain, by the principles detailed in the preceding 
articles, a preponderating evidence in favor of that boundary. 

Art. 28. By the mere mental substitution of arcs of great 
circles for straight lines, the whole preceding theory, with 
scarcely a verbal alteration, becomes directly applicable to all 
cases, in which, from the large extent of the total tract, we 
may suppose the rotundity of the terrestrial surface to have 
any perceptible effect on the question. 

Art. 29. But, in the practical application of this theory, 
the following indispensable precaution must be observed. 
Through all the preceding investigations, the extent of any 
one boundary is conceived to be that wherein it separates, 
every where, the previovsly undetermined shares of those two par- 
ties, and of those two parties alone. And therefore, if a bounda- 
ry be part of a continuous frontier ; and if that frontier, in 
other parts of its course, be a boundary to shares predetermin- 
ed between the same two parties, or a boundary at all apper- 
taining to any share of any third party ; if we would apply 
the preceding investigation to this first boundary, we must 
first ascertain hoio mucli of that frontier constitutes this boun- 
dary, by discovering on what points of that frontier the extrem- 
ities of this boundary fall. 

Art. 30. Hence, in the application of the preceding process 
to an examination of the two boundaries respectively claimed 
by the American and British nations, on the frontiers of New 
Brunswick and Maine ; we should first ascertain where the 
common ivestward extremity of the tico covficting boundaries 
lies ; that is, to what extent westward, in 1783, the boundary 
established in that year separated British from American ter- 
ritory. The true answer to this question seems to be, that 
such boundary was bona fide a boundary between the United 
States and the British possessions, through all its westward 
course till it first reached a branch of the Mississippi ; as every 
part of the region westward of that river, at least as far as to 
the Rocky Mountains, appears to have been, at that time, ei- 
ther Indian or French territory. This question, however, be- 
ing subject to the decision of historical research, its further 
consideration is unnecessary here. 



20 



AN EXAMINATION OF THE TREATY OF 1783 : 

FROM WHICH IS DERIVED 

A DEMONSTRATION OF THE FALSEHOOD OF 
THE BRITISH CLAIM, 

AND ALSO OF 

THE TRUTH OF THE AMERICAN. 



Art. 1. An angle of any superficial figure, lies at a point 
where two sides of the figure meet, as the point A in Fig. 16. 

Art. 2. Two sides of any such figure can meet only at their 
extremities. 

Art, 3. Hence the angles of any such figure can exist only 
at the extremities of the several sides. 

Art. 4. Hence, any such figure has no angle at a point on 
any of its own sides, between the extremities of that side. 

Art. 5. It may easily, however, have an angle of another 
figure, or several angles of as many several figures, at such a 
point on its own side, between the extremities of that side ; as 
the angles a and c, at the point Z», in Fig. 17, where AB, BC, 
CD, and DE, are sides of a figure whereof the total repre- 
sentation is not necessary here ; and h d is u. common side of 
two other figures, whose representation is also unnecessary. 

Art. 6. Before, during, and after the preparation of the 
treaty of 1783, for a certain space of time, the two tracts of 
country, which are now called Nova Scotia and New Bruns- 
wick, were comprehended under the common appellation of 



21 

Nova Scotia. See the article, ^' Nova Scotia,'' in the En- 
cyclopedia Americana, and also in Dr. Rees's Cyclopedia. 

Art. 7. In that comprehensive sense, we shall employ the 
term in this examination. 

Art. 8. Now it appears from the map of Lower Canada, 
with adjacent parts of the United States and Nova Scotia, pub- 
lished by the British " Society for the Diffusion of Useful 
Knowledge," that ojie side of Nova Scotia, namely, the west- 
ern, is a line extending North and South, as BC in Fig. 17 ; 
whose northern extremity C meets another side, namely, CD, 
and its southern extremity meets the Chiputnaticook river, 
which, by the line of its progress to Passamaquoddy Bay, 
makes a third side. It also appears from that map, that, ac- 
cording to the British claim, the " northwest angle of Nova 
Scotia,'' specified in the treaty of 1783, lies on the point b, 
of the side BC, between the extremities of that side, and b d 
is a common side or boundary of the United States and of 
Canada. But, by what we have shown in articles 4 and 5, 
the angle a would, in that case, be the southeast angle of Can- 
ada, and the angle c would be the northeast angle of the Uni- 
ted States ; but Nova Scotia has no angle whatever at the 
'point b ; therefore the British claim must be false. 

Art. 9. But, by the American claim, the northwest angle 
of Nova Scotia lies at the point C, where undoubtedly there 
is an angle of Nova Scotia; which angle, being also the only 
angle of Nova Scotia which is made by the northern extrenii- 
ty of the side BC; ii must be the northwest angle of Nova 
Scotia, specified in the treaty aforesaid. Therefore, the Brit- 
ish claim is false, and also the American is true. Q. 
E. D. 

I cannot conclude the foregoing investigations, without ac- 
knowledging my obligations to Dr. J. D. Hedge, of Cam- 
bridge, for the kind and active manner in which he has labored 
to ensure the correctness, encourage the publication, and pro- 
mote the success of this work, by the removal of many prac- 



22 

tical difficulties, which, otherwise, if not insurmountable, 
would have created extreme delay and embarrassment ; more 
especially I refer to the exertions of that gentleman for the 
facilitation of my access to authorities and documents, which 
I should otherwise have _found it extremely difficult, if not 
impossible, to procure. 

Finally, Sir, having now completed a task, in the perform- 
ance of which I have been animated only by an impulse of 
duty, combined with an encouraging hope of inheriting a share 
of that immortal celebrity, which the man who benefits his 
race by intellectual achievements may extort from the grasp 
of an unwilling world, and retain to the latest extent of hu- 
man posterity ; 

I subscribe myself, 

with great respect, 

your Excellency's 
obedient servant, 

JOHN LEE. 



fi-0 




.-T% 



BB 



9. 



3."* 









^. 



^o^ 








y^ff^^r 



»>^# 



1 o,* 






>^ ,.. 



W5 






^K^^ 



'.^/% 



^.'^^ <-^" 



,<^^- 






-^^ 



K^ 












:f 



A 



.%■ 









% 



1*0, 



> 



ST USTINE 



LIBRARY OF CONGRESS 



011 895 611 4 



